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## What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with punctures on it, or torus with a cusp attached to it . They could also be torus torus with one or more than one handles attached to it.

For example, see the diagrams on : http://www.maths.bris.ac.uk/~mazag/hyperbolic/index.html

to get ideas about what surfaces I am talking about. They are not given by any easy equations.

Is there a software I can use to draw them ? People who study Riemann Surfaces or Hyperbolic Geometry or Teichmmuller Theory would definitely know exactly what surfaces I am talking about.Please let me know if you use such a software. Thanks a lot in advance !!

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I think this is off-topic, so I have voted to close. Any decent drawing program will do the job. I personally use inkscape. – Andy Putman May 10 2012 at 22:48
A hardware solution consisting of a human, a piece of paper, a pencil, and a scanner works wonders. – Igor Rivin May 10 2012 at 23:43
This was cross-posted at TeX-SX: tex.stackexchange.com/q/55326/86 where it was linked to this question: tex.stackexchange.com/q/17031/86 which led to a TikZ/PGF package for drawing cobordism diagrams. – Andrew Stacey May 11 2012 at 7:01
@ Igor Rivin : I would very happily draw the diagram pretty neatly by my hand and incorporate in the article that I am writing, but do the journals accept hand-drawn pictures in the papers ? I guess I might just ask them. Although, I don't see there would be a way one could be able to distinguish ! – Analysis Now May 11 2012 at 16:20

The solutions that people have mentioned are good if you are happy with a two-dimensional line drawing. It would be better to have three-dimensional equations that could be plotted using Maple, or something like that, but that seems to be surprisingly hard. The equation $$3x_3^2x_4-2(x_1^2+x_2^2)x_4-2x_4^3+2(x_1^2-x_2^2)x_3 = 0$$ defines a highly symmetric surface of genus 2 embedded in $S^3$, and one can project stereographically into $\mathbb{R}^3$ to get a nice picture like this:

(There's a lot to be said about this example; I will have an undergraduate working on it over the summer.) However, I do not know similarly nice equations for surfaces of higher genus, or with the two tori in the same plane rather than at right angles, or with cusps.

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Very nice, but for illustration purposes you can certainly just construct a surface out of the usual two-hold tori with your favorite CAD software suite. – Igor Rivin May 11 2012 at 11:53
@Igor: have you actually done that? I made the pictures at neil-strickland.staff.shef.ac.uk/talks/durham.pdf using Blender, and found it extremely painful to produce anything that looked smooth and natural. – Neil Strickland May 11 2012 at 12:20
@Neil Those are nice images. But please have a look at the 3D-XplorMath Mathematical Visualization program and the Virtual Math Museum mentioned in my answer below. We have worked hard to provide high quality images for use in articles such as yours and also the tools for creating them easily, precisely to avoid people having to continually reinvent the wheel and the resulting "extremely painful" experiences you mention. – Dick Palais May 11 2012 at 13:08
@Dick: thanks for the pointer. The pain arises when you want to build a 3D digital model from a mental image or 2D sketch, and you do not have equations of any kind. You can start with some cylinders or tori, say, but then you need to stitch them together and smooth out the join. I could not see anything in the 3D-XplorMath that would help with that - am I missing something? People routinely do much more complicated things with Blender (similar to what you see in animated movies) so presumably it is easy if you are practised and/or talented, but I found it hard. – Neil Strickland May 11 2012 at 14:44
@Dick: I could be wrong, but it certainly looks to me as though the questioner only sketches to work with, and not any kind of implicit or explicit equations. – Neil Strickland May 11 2012 at 15:52
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If you go to the Algebraic Surface page of The Virtual Math Museum at

http://virtualmathmuseum.org/Surface/gallery_o.html#AlgebraicSurfaces

you will see many nice examples. These were created using the program 3D-XplorMath which you can download at http://3D-XplorMath.org . If you install that and go to the Implicit Surface category you will see many examples with documentation.

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